Finite Group Databasebeta

← Prev (#1225) Next (#1227) →

Cayley diagram

D3×AGL(1,𝔽5) #1226

order120
isoclinismisoclinic class of D3×AGL(1,𝔽5)
automorphismAut(G) = D3×AGL(1,𝔽5) (order 120) · Inn(G) = G · Out(G) = C1
holomorphG ⋊ Aut(G) = D32×AGL(1,𝔽5)2
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productD3×AGL(1,𝔽5)
unambiguous semidirect product(C3×AGL(1,𝔽5))⋊C2
unambiguous semidirect productΓL(1,𝔽16)⋊C2

Splittings

Fitting, socle, derived, HallC3×C5 ⋊ C2×C4
SylowC5 ⋊ C4×D3
SylowC3 ⋊ C2×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5C2×C4
prime radicalD3×D5C2
cosocleC3×D5C22
LCSG ▷ C3×C5
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 15

A minimal faithful permutation representation — 2 generators on 8 points:

(2,3,4,5)(6,7,8);(1,2,3,5,4)(7,8)

Conjugacy classes & character table

15 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×AGL(1,𝔽5), D9×AGL(1,𝔽5), C3×D3×AGL(1,𝔽5), …
maximal normal subgroup ofC2×D3×AGL(1,𝔽5), C3×D3×AGL(1,𝔽5), C5×D3×AGL(1,𝔽5), …
maximal quotient ofDic3×AGL(1,𝔽5), (C2×D3)⋊AGL(1,𝔽5), Dic3⋊AGL(1,𝔽5), …
automorphism group ofD15, ΓL(1,𝔽16), D3×D5, …
holomorph ofC3×C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 36);